# Paper 2, Section II, A

(a) Find the general solution of the system of differential equations

$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)$

(b) Depending on the parameter $\lambda \in \mathbb{R}$, find the general solution of the system of differential equations

$\left(\begin{array}{l} \dot{x} \\ \dot{y} \\ \dot{z} \end{array}\right)=\left(\begin{array}{rrr} -1 & 2 & -1 \\ 1 & 0 & -1 \\ 1 & -2 & 1 \end{array}\right)\left(\begin{array}{l} x \\ y \\ z \end{array}\right)+2\left(\begin{array}{r} -\lambda \\ 1 \\ \lambda \end{array}\right) e^{2 t},$

and explain why $(2)$ has a particular solution of the form $\mathbf{c} e^{2 t}$ with constant vector $\mathbf{c} \in \mathbb{R}^{3}$ for $\lambda=1$ but not for $\lambda \neq 1$.

[Hint: decompose $\left(\begin{array}{c}-\lambda \\ 1 \\ \lambda\end{array}\right)$ in terms of the eigenbasis of the matrix in (1).]

(c) For $\lambda=-1$, find the solution of (2) which goes through the point $(0,1,0)$ at $t=0$.